Subtract the following rational expressions. $\dfrac{7x-8}{9x+1}-\dfrac{1}{5x^3}=$
Solution: We can subtract two rational expressions whose denominators are equal by subtracting the numerators and keeping the denominator the same. [Does this fit with how we subtract rational numbers?] When the denominators are not the same, we must manipulate them so that they become the same. In other words, we must find a common denominator. Since the two denominators do not share any common factors, the common denominator is simply the product of these two denominators: $({9x+1})\cdot({5x^3})$. Let's manipulate the expressions to have that denominator: $\begin{aligned} &\phantom{=}\dfrac{7x-8}{{9x+1}}-\dfrac{1}{{5x^3}} \\\\ &=\dfrac{(7x-8)\cdot({5x^3})}{({9x+1})\cdot({5x^3})}-\dfrac{1\cdot({9x+1})}{({5x^3})\cdot{9x+1})} \end{aligned}$ [Why did we do that?] Now that both denominators are the same, let's subtract! $\begin{aligned} &\phantom{=}\dfrac{(7x-8)\cdot(5x^3)}{(9x+1)\cdot(5x^3)}-\dfrac{1\cdot(9x+1)}{(5x^3)\cdot(9x+1)} \\\\ &=\dfrac{(7x-8)\cdot(5x^3)-1\cdot(9x+1)}{(9x+1)(5x^3)} \\\\ &=\dfrac{35x^4-40x^3-9x-1}{(9x+1)(5x^3)} \end{aligned}$ In conclusion, $\dfrac{7x-8}{9x+1}-\dfrac{1}{5x^3}=\dfrac{35x^4-40x^3-9x-1}{(9x+1)(5x^3)}$